Sufficent condition for existence of power series of Real valued function

Question

I know that for complex valued function if function is holomorphic on some open connected set then there exist convergent power series converging to that function.
But this is not true in case of infinite time differentible Real valued function .Example $e^{-1/x^2}$
I have not encountered yet Sufficent condition for existence of Power series for real valued function.?
I had encountered taylor series but There is residue term is present.So how to say Taylor series as power series of function? Any Help will be appreciated .

Answer 1

There is a simple criterion to conclude the convergence of the power series in some point $x \in I$ towards the function, that is to show that the remainder term in the Taylor approximation vanishes on a neighborhood of $x$. For example, we could use special remainder estimates as the Peano form.

In general, you can't say more. For example, you can construct smooth functions which are nowhere analytic, see here.

Your question can be modified by asking if all derivates in one point already determine the function uniquely. This question is more general when the question of convergence. (In fact, it can happen that the Taylor series doesn't convergence, but the function is uniquely determined by the derivates in this point.)

This class of functions are called quasi-analytic, see here for a full description, and there is a simple criterion, that is the Denjoy–Carleman theorem.